The definition of the lowlink in Tarjan's algorithm is the
smallest index of a vertex that is reachable with at most one
back-edge in SCC.  This is not useful for a cross-edge.

If we start traversing from A in the following graph, the final
lowlink of D is 3.  The cross-edge here is one between D and C.

  A -> B -> D   D = (4, 3)  (index, lowlink)
  ^    |    |   C = (3, 1)
  |    V    |   B = (2, 1)
  `--- C <--'   A = (1, 1)

This is because the lowlink of D is updated with the index of C.

In the following patch, we detect a dead SCC by checking two
conditions for each vertex.

  1) vertex has no edge directed to another SCC (no bridge)
  2) vertex's out_degree is the same as the refcount of its file

If 1) is false, there is a receiver of all fds of the SCC and
its ancestor SCC.

To evaluate 1), we need to assign a unique index to each SCC and
assign it to all vertices in the SCC.

This patch changes the lowlink update logic for cross-edge so
that in the example above, the lowlink of D is updated with the
lowlink of C.

  A -> B -> D   D = (4, 1)  (index, lowlink)
  ^    |    |   C = (3, 1)
  |    V    |   B = (2, 1)
  `--- C <--'   A = (1, 1)

Then, all vertices in the same SCC have the same lowlink, and we
can quickly find the bridge connecting to different SCC if exists.

However, it is no longer called lowlink, so we rename it to
scc_index.  (It's sometimes called lowpoint.)

Also, we add a global variable to hold the last index used in DFS
so that we do not reset the initial index in each DFS.

This patch can be squashed to the SCC detection patch but is
split deliberately for anyone wondering why lowlink is not used
as used in the original Tarjan's algorithm and many reference
implementations.
Signed-off-by: Kuniyuki Iwashima <kuniyu@amazon.com>
Acked-by: Paolo Abeni <pabeni@redhat.com>
Link: https://lore.kernel.org/r/20240325202425.60930-13-kuniyu@amazon.comSigned-off-by: Jakub Kicinski <kuba@kernel.org>